Concepts and Contracepts

almost immediately after posting about using gppf as a measure of roundness, another one of course crossed my mind. i've been calling it IFR, for Inner Factor Ratio. by "inner factor"(s), i mean the two factors closest to the number's square root. so, for instance, the factors of 12 are 1, 2, 3, 4, 6, 12. the two numbers closest to its square root are 3 and 4, so its IFR is 4/3.

i came to this measure while working on the gppf stuff, actually. making lists of factors of a number, i noticed that some numbers had lists with really large gaps in the middle (e.g. 57: 1, 3, 19, 57, the gap being between 3 and 19), and some didn't (e.g. 56: 1, 2, 4, 7, 8, 14, 28, 56, the gap being between 7 and 8), and that the round numbers tended to be the ones with small gaps. (remember that, as we're working multiplicatively, the "middle" of a list like this is necessarily the square root of the number.) of course (again with the multiplicative), it makes more sense to measure the size of this gap using a ratio instead of a difference. thus, Inner Factor Ratio.

an obvious concern is whether the IFR is well-defined for squares. if you consider something like 36: 1, 2, 3, 4, 6, 9, 12, 18, 36, the middle/square root is 6, which is actually in our list. so do we want the ratio of 6 and 4, or the ratio of 9 and 6? conveniently, these two ratios will always be the same. ( Collapse ). so the IFR is even well-defined in this situation.

by definition, the IFR is always greater than 1 (it being the ratio of two numbers, with the bigger one on top), and less than or equal to n (which upper bound is attained if n is a prime number). one could thus normalize the IFR by dividing by n, or by taking the log_n, but i'm disinclined to do this. in at least some cases, the IFR has some nice scaling properties. IFR(k²n) ≤ IFR(n) because if a, b, are the inner factors of n, then ak, bk are candidates for the inner factors of k²n. we also note that if p is prime, the factors of pⁿ are all the intermediate powers of p, and thus the IFR of pⁿ for any n is necessarily p. so in some sense the IFR is scale-insensitive.

based on those results, it's tempting to make what i call the strong domination conjecture: IFR(mn) ≤ min(IFR(m),IFR(n)). this is false. consider, e.g., 606 = 6 * 101. IFR(6) = 3/2 = 1.5. but IFR(606) = 101/6 = 16.8333 is significantly larger than 1.5. a nerfed version, however, the weak domination conjecture IFR(m*n) ≤ max(IFR(m),IFR(n)) is true. ( Collapse )

the asymptotics of IFR are still somewhat vague. i've tabulated it up to 1753 so far. mean values will be dominated by primes and biprimes, so i've looked at medians instead. the median value of IFR(n) seems to stay fairly consistently around n^0.26, but, as i say, that's still somewhat vague. minimal values are attained by numbers in the form n(n+1) (A002378). somewhat more problematic, lurking close to these are some biprimes, specifically those that are products of twin primes. my æsthetic has a problem calling a biprime "round", even one with a nice IFR like 41 × 43 = 1763.

of course, i'm posting about this now because just this morning, thinking about this biprime issue lead me to yet another way of measuring roundness, which i like better! but i'm gonna play with it a bit before writing it up.

because i never really stop thinking about anything, i'm going to again go on for a little bit about quantifying the roundness of numbers. way back here, i mused about looking at the greatest prime factor gpf(n) of an integer, and defining an integer's roundness as log_ngpf(n). at the time, i commented:

the problem with this method is that it tends to overprivilege prime powers. any number of the form n = p^k for p prime will end up measuring as very round in this calculation. in fact, this calculation establishes them as the archetypal round numbers. which feels a little off to me: a round number should play nicely with more than one prime.

for some reason, it took three and a half years to see the obvious way around that. (which raises the obvious question: what do i mean by "obvious"?) instead of looking at the greatest prime factor of n, look at the greatest prime power factor of n. gppf(n) has some curious properties that gpf(n) doesn't. it's an ultimately increasing function, for instance: for any n, there's a threashold N_n such that m > N_n means gppf(m) > n. isn't that nifty?
i've been tabulating values for a few days now (thanks to A034699 for getting me started), and still don't have a really good handle on its asymptotic behaviour. but it looks like a fascinating approach. notably round numbers (i.e. ones that have a lower roundness index than anything before) so far are 6, 12, 20, 30 (the first one with gppf(n) less than the square root of n), 60, 210, 420 (the highest number with a gppf of 7), 840 (the highest number with a gppf of 8), 1260.
i've only tabulated to 1763 so far (i'm doing it partly by hand, using a moderately buggy smidge of common lisp code gakked from here), but the next time i get the proper alignment of free time and will, will likely throw maple at the problem. i'm expecting good things from 2520 and 27720, for "obvious" reasons.

both π day and the more recent τ day (both more recent = it was yesterday and more recent = it hasn't been celebrated for as many years) suffer from the same problem: they depend on the decimal expansion of one or t'other of the circle constants. using base 10 is precisely the sort of arbitrary choice that going to radian measure of angles is attempting to avoid. thus, one wonders, is there a more appropriate way to select a day to celebrate all things circular?

one obvious choice would be to celebrate when the earth has moved through one radian of its orbit, i.e., when the distance it's travelled (from, e.g., perihelion) is equal to its radius from the sun. there are far fewer mathematically arbitrary choices involved in this. perihelion is around january 4. and 365.242199/τ (365.242199/(2π) for the π-ous) is 58.130101389 = 58 days + 3 hours + 7 minutes + 20.76 seconds. doing the math suggests that radian day would be around march 4th, which conveniently is also GM's day.

two problems:

1. someone else thought of it first. technically, this is the opposite of a problem. although, by starting the radian-clock rom perihelion (i suppose one also might want to use winter solstice?), my radian-day ends up on a slightly different date.
   
   
2. this one, however, is a problem: the earth's orbit famously ain't a circle. thank you, johannes kepler! it's an ellipse. and calculating lengths along an ellipse will require futzing about with, well, elliptic integrals (guess why they're called that). maybe later.

jugglingstore, mian1wzd, neverexost, and renegadesufi, i'm curious about how you came to friend me on LJ. where did you find my journal? you realize that, as per my friending policy, discussed on my profile page, i'm not gonna be friending you back any time soon. right?

just wondering. this prolly reads snarkier than it should. you're of course welcome to hang out. just don't expect much of interest, except for math. pretty much everything else is locked down.

ever since first seeing walken dance his way through the video for Fat Boy Slim's Weapon of Choice it's been obvious to me that the song needed a mathematical filking. thus, i present

Axiom of Choice

Don't be shocked by Cohen's and Gödel's voice
Check out my new axiom, axiom of choice
Don't be shocked by Cohen's and Gödel's voice
Check out my new axiom, axiom of choice, yeah
Listen to Cohen's and Gödel's voice (aah...)
You can check it on out, it's the axiom of choice, yeah

You well-order this this
Or you well-order that
You well-order this
Or you well-order that
Or you well-order this
Or you well-order that
Or you well-order us

You well-order this
Or you well-order that
You well-order this
Or you well-order that
You well-order...

Principle of Hausdorf, a lemma due to Zorn
Principle of Hausdorf, and a lemma due to Zorn
Principle of Hausdorf, and a lemma due to Zorn
With a Principle of Hausdorf, uh, we're gettin' warm

Ambiguously known?
As constructivists, they moan
At pure existence proofs
and my pal
I guess you just don't realize
It's gone beyond doublin' size
We're mathematical sleuths
For new truths

some parts of this are clearly stronger than others. suggestions for improvement are very much welcome.

i note also that the original material has several other lyrics, on loops, i think not performed by the main vocalist. ( Collapse )

on a walk the other day, i noticed that several of the passing (three-digit) license plate numbers i factored ended up having a greatest prime factor (gpf) of 31. as in, i think, four in six consecutive plates. of course, i wondered if there was anything to that. as the largest three-digit number is 999 (roughly 1000), and the square root of 1000 is roughly 31, i immediately conjectured that the average greatest prime factor around the number n should be roughly the square root of n. stated slightly more formally: if one defines a function f of x to be the sum of gpf(n) for all n between 1 and x (what is the gpf of 1? 0? 1?), then either f(n)/n or f′(n) is roughly the square root of n. equivalently, i conjectured that f(n) is O(n^1.5)

conveniently, the OEIS page for the gpf sequence links to the first 100000 gpf values. slapping those in a handy dandy spreadsheet and doing the relevant calculations suggests that (at least as far as i've run the calculations so far) f is more like O(n^1.76) or even higher. i'm starting to suspect that it might be O(n^?log n) for some value of ?, but haven't bothered playing around with that yet.

for all i know, someone's already figured this out. one of these years, i'll have to figure out how to search the literature.

(oh, also? the most common gpf for numbers between 1 and 999 is 7, nowhere near 31. yesterday's walk-observation was almost certainly just ramsey theory messing with me.)

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why is it that the african nation called the gambia is called "the gambia" instead of just "gambia"? i recognize that articles on country names are common in other languages. auf franzözisch, zum beispiel, man sagt "le canada", aber auf englisch man sagt nicht "the canada". i remember that we used to call ukraine "the ukraine", but that stopped somewhere along the way. so any guesses on the gambian (or should i say "the the gambian") situation?

i continue to think about roundness quantification when i've got nothing better to do. another approach, one might even say a classic approach, would be to ask how many numbers smaller than n share factors with n. if there are lots of them, n is round. not so many, not so round.
quantifying this is pretty easy, thanks to Euler's totient function φ(n). you'll recall that φ(n) is the number of numbers smaller than n which are relatively prime to n (i.e. they have no common factors with n other than 1). so if φ(n) is small, n is round; if φ(n) is big, n is not. except that "small" and "big" are relative to the size of n itself, so we're better off normalizing it: φ(n)/n.
of course, there's a formula for φ(n) based on the prime factorization of n. if n=Πp_i^k_i, then φ(n) = n*Π(1-1/p_i) (products taken over all prime factors of n). normalizing this gives φ(n)/n = Π(1-1/p_i). easy peasy.
one reason i'd previously dismissed this measure is that it fails to reward repeated prime factors. 6 and 12 and 18 and 24 and 36 are all divisible by 2 and 3, and thus all end up with a normalized totient of (1 - ½)(1 - ⅔) = ⅓ — nice and low, to be sure, but all give the same result. but i'm starting to warm to that idea. is 20 really much rounder than 10?

in any case, in this measure, maximal values are given by primes (and, of course, given the above, prime powers), with normalized totient values of (p-1)/p, tending to 1. minimal values are given by primorials, and i believe (but haven't yet sorted out a proof, so i might be worng) their normalized totients will tend to zero.

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